Neil Tennant has made what seems to me the following excellent point:
"But this misses the philosophical point of my earlier mailing about
Fefereman's "gut feeling" that CH is "inherently vague". My challenge was
to have someone show why the *component expressions* in CH had indefinite
*senses*. I was pointing out that one cannot have one's cake and eat it.
If those expressions have definite enough senses for at least *some*
sentences involving them to have definite truth-values, then why should it
be the case that a sentence like CH, involving no new expressions, is
itself suddenly "vague" or "indefinite" in sense, and thereby deprived of
a determinate truth-value?"
To put the matter another way, CH is stated in a language that
mathematicians use all the time. If indeed it does not express a
"definite proposition", then it would seem a matter of urgency to draw
a line between the sentences of this language which do express such
propositions and those which don't, and to offer some theory of meaning
which justifies this line. Absent such a systematic approach, the
assertion that CH has no definite meaning amounts to the assertion
" I don't understand its meaning".
Torzel Franzen suggests:
What is indefinite (on the view at issue) is the range of the quantifiers:
"for every set of reals there is a mapping...".
and Jerry Seligman gives a homespun example of a statement whose vagueness
results from quantification over a domain (the inhabitants of a town)
with vague boundaries. Unfortunately, the example seems too far from the
set theory case to help: whether or not a given object is a set of reals
does not seem to admit borderline cases. In any case, if the quantifier
over sets of reals (and even the quantifier over reals, according to
Feferman) can contribute vagueness, what about statements like
1. There is a non-Lebesgue measurable set
2. There is a nonprincipal ultrafilter on omega
or, in the case of the real quantifier
3. Every analytic set is Lebesgue measurable
4. Every open game is determined
5. Every Borel game is determined
6. Every analytic game is determined
Which of these sentences express a definite proposition?
If one rejects the constructivist retreats to less expressive
languages (perhaps because they toss out good mathematics in order to
save inherently vague philosophy), I see no principled line to draw
between CH and any other statement of 3rd order arithmetic.
That doesn't mean there is no such line. In this connection, Franzen
says
"I don't think it's very rewarding to try to squeeze any precise or
theoretical content out of this notion of "inherent vagueness". Its main
thrust is pragmatical: there is no reason to believe that our mathematical
intuition settles every question, and CH seems a case in point."
I don't entirely agree here. There is a reason that CH has been the
holy grail of set theory since its birth. It is an obviously important
question at the foundations of the subject. Indeed, any bright high school
student can see that in a reasonably short time. It is important for the
foundations of set theory, and hence mathematics, to decide it-- or to
show it is undecidable (perhaps because it is vague). Whichever way it
turns out, true, false, meaningless, or some blend of the three, I suspect
the solution will involve some philosophical/conceptual analysis of what
it is to be a solution.
A solution to the Continuum Problem will almost certainly also involve
some metamathematical sophistication involving models of set theory. In
this respect it is indeed different from the Riemann Hypothesis
(probably), as Feferman points out. To one whose focus is foam, this makes
CH more interesting than the Riemann Hypothesis. As Lee Stanley says, we
are still "gathering data" about models of set theory. However, some
broad patterns have emerged. These suggest to me (and, I believe, some
other set theorists) that one might argue that one can solve the Continuum
Problem by finding a natural, generically absolute theory of P(R) which
is consistent with all our large cardinal axioms. There is more to say
here than can fit in the margin of this note, and I certainly would not
claim the "argument" is airtight. My point at the moment is just that
we shouldn't waste time hankering after the days when the apple would
hit you on the head, we should go after the problem with all the tools we
have.
John Steel